Definition 3.11 (Blahut-Arimoto algorithm [1]}, [2]}) 
Let \(X\)  be a random variable on the discrete space of locations \(\mathcal {X}\)  with a PMF \(\pi _{\mathcal {X}}\) . For a metric \(d\colon \mathcal {X}\times \mathcal {X}\mapsto \mathbb {R}_{\ge 0}\)  and a fixed \(d^*\in \mathbb {R}^+\) , we wish to find a channel \(\hat{\mathcal {C}}\in *{C}(\mathcal {X},\mathcal {X})\)  giving \(RD(X,d^*)\) . In other words, we wish to find a channel \(\hat{\mathcal {C}}\)  such that:
\(\hat{\mathcal {C}}=\operatornamewithlimits{arg\,min}\limits _{\begin{array}{c}\mathcal {C}\in *{C}(\mathcal {X},\mathcal {X})\\AvgD(X,\mathcal {C},d)\le d^*\end{array}}{MI(X,Y_{X,\mathcal {C}})}\) 
